Using (1a)-(1f), the aerodynamical roll, pitch, and heading moments and lift force, which are induced by the reference flight trajectory, are expressed as follows:
where the term [[??].sup.2] is used to minimize the induced aerodynamical heading moment and the second term [mathematical expression not reproducible] is introduced in the proposed cost function to minimize the induced efforts and consumed energy.
Figures 10 and 11 illustrate the induced lift force, aerodynamical moments, and their mean values computations with respect to each scenario.
Based on differential flatness approach, the quadcopter UAV dynamical constraints were checked instantaneously by computing the induced aerodynamical roll, pitch, and heading moments and lift force, respectively.
Caption: Figure 6: Induced efforts: (a) lift force and aerodynamical (b) roll and (c) pitch and (d) heading moments, respectively.
Caption: Figure 7: Mean values of (a) lift force and aerodynamical (b) roll and (c) pitch and (d) heading moments, respectively.
The maximum, minimum, and mean values of the vehicle aerodynamical coefficients during the overtaking maneuvers are shown in Tables 10 and 11.
According to the optimizing rate, this method can effectively optimize all the bus aerodynamical coefficients during the overtaking maneuvers.
For the car aerodynamical coefficients, this method can effectively optimize [C.sub.d], [C.sub.s], [C.sub.l] of car, while the optimizing effects of [C.sub.pm] and [C.sub.m] are not satisfactory.
Length (m) 12.000 Width (m) 2.500 Height (m) 3.400 Frontal Area ([m.sup.2]) 9.300 Side Area ([m.sup.2]) 45.600 Wheel base between front and back axles (m) 6.400 Radius of the cylinder at the upper leading edge of tail (m) 0.300 Radius of the cylinder at the lower leading edge of tail (m) 0.100 [alpha] in Figure 1 (degree) 90 [beta] in Figure 1 (degree) 80.9 Length of the upper cylinder (m) 1.900 Length of the lower cylinder (m) 2.300 Table 2: Aerodynamical coefficients in different blockage ratio.
Newton's aerodynamical problem consists in determining the minimum resistance profile of a body of revolution moving at constant speed in a rare medium of equally spaced particles that do not interact with each other.
More precisely, we considered the 1696 brachistochrone problem (B); the 1687 Newton's aerodynamical problem of minimal resistance (N); a recent brachistochrone problem with restrictions (R) studied by Ramm in 1999, and where some open questions still remain ; and finally a generalized aerodynamical minimum resistance problem with non-parallel flux of particles (P), Recently studied by Plakhov and Torres [11;14] and which gives rise to other interesting questions .